Find the Inflection Points f(x)=x^3-12x

Problem

ƒ(x)=x3−12*x

Solution

1. Find the first derivative by applying the power rule to each term of the function.

d(ƒ(x))/d(x)=3*x2−12

2. Find the second derivative by differentiating the first derivative with respect to x

d2(ƒ(x))/(d(x)2)=6*x

3. Set the second derivative to zero to find potential inflection points where the concavity might change.

6*x=0

4. Solve for x to determine the candidate for the inflection point.

x=0

5. Test the concavity by checking the sign of the second derivative on intervals around x=0 For x<0 ƒ(x)″<0 (concave down), and for x>0 ƒ(x)″>0 (concave up). Since the concavity changes, x=0 is an inflection point.

6. Find the y-coordinate by substituting x=0 back into the original function ƒ(x)

ƒ(0)=(0)3−12*(0)=0

Final Answer

Inflection Point=(0,0)

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